[Paper Review] Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds

David Glickenstein|arXiv (Cornell University)|Jun 8, 2009

Geometric Analysis and Curvature Flows30 references4 citations

TL;DR

This paper introduces a unified framework for discrete conformal variations on piecewise flat 2D and 3D manifolds, enabling explicit computation of angle and curvature variations under conformal deformations. The key contribution is a geometric derivation of variation formulas that generalize prior results, leading to convexity and rigidity theorems for curvature functionals, including Einstein-Hilbert-Regge functionals, with applications to scalar curvature convergence and critical metric classification.

ABSTRACT

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This variation generalizes variations within the class of circles with fixed intersection angles (such as circle packings) as well as other formulations of conformal variation of piecewise flat manifolds previously suggested. We describe the angle derivatives of the angles in two and three dimensional piecewise flat manifolds, giving rise to formulas for the derivatives of curvatures. The formulas for derivatives of curvature resemble the formulas for the change of scalar curvature under a conformal variation of Riemannian metric. They allow us to explicitly describe the variation of certain curvature functionals, including Regge's formulation of the Einstein-Hilbert functional (total scalar curvature), and to consider convexity of these functionals. They also allow us to prove rigidity theorems for certain analogues of constant curvature and Einstein manifolds in the piecewise flat setting.

Motivation & Objective

• To develop a general framework for conformal variations on piecewise flat manifolds that unifies disparate notions of discrete conformality in 2D and 3D.
• To derive geometric formulas for the variation of angles under conformal deformations, avoiding reliance on explicit algebraic computation.
• To establish convexity and rigidity properties of curvature functionals, particularly the Einstein-Hilbert-Regge functional, under conformal variations.
• To extend results on scalar curvature convergence and critical metrics (e.g., Ricci flat, constant scalar curvature) to the discrete setting.
• To provide a foundation for proving discrete analogues of smooth Riemannian theorems, such as Obata’s rigidity result.

Proposed method

• Introduces a new definition of conformal variation on piecewise flat manifolds using edge length scaling via vertex functions.
• Derives variation formulas for angles in both 2D and 3D using geometric arguments, generalizing prior explicit computations.
• Applies these formulas to curvature functionals, particularly the Einstein-Hilbert-Regge functional, and computes first and second variations.
• Uses discrete Laplacians and matrix definiteness conditions (e.g., from Proposition 39) to analyze convexity of functionals.
• Establishes convexity of the functional under specific geometric constraints, such as non-negative vertex curvatures or positive edge-length adjustments.
• Applies the theory to prove rigidity theorems for zero-curvature and Ricci-flat metrics under conformal variations.

Experimental results

Research questions

• RQ1 How can conformal variations be consistently defined on piecewise flat 2D and 3D manifolds to enable curvature analysis?
• RQ2 What geometric conditions ensure the convexity of curvature functionals like the Einstein-Hilbert-Regge functional under conformal variations?
• RQ3 How do variations of angles in piecewise flat manifolds relate to underlying geometric quantities such as areas and edge lengths?
• RQ4 Under what conditions is a piecewise flat manifold rigid under conformal variations, particularly for zero or Ricci-flat metrics?
• RQ5 Can discrete analogues of smooth Riemannian theorems—such as Obata’s rigidity result—be established in the piecewise flat setting?

Key findings

• The paper proves that the Einstein-Hilbert-Regge functional is convex on 3D piecewise flat manifolds when all vertex curvatures are non-negative and edge-length adjustments satisfy a positivity condition.
• A two-dimensional piecewise flat manifold with zero vertex curvature is rigid under conformal variations if it satisfies any of the conditions (1)–(5) in Proposition 39.
• A three-dimensional piecewise flat manifold that is Ricci flat is rigid under conformal variations if it satisfies condition (1) or (6) in Proposition 39, analogous to Obata’s theorem in the smooth case.
• The variation of angles in piecewise flat manifolds is geometrically interpretable and computable via a unified framework, generalizing prior explicit formulas.
• The theory provides a systematic way to analyze curvature convergence and critical metrics in discrete geometry, with implications for Alexandrov’s theorem and discrete Einstein manifolds.
• The framework unifies existing approaches such as circle packing, sphere packing, and face-based conformal structures under a single variational formalism.